Distribution of maximum of random variables taken over an uncountable set

Asked Aug 22 '23 at 09:32

Active Aug 22 '23 at 14:14

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Consider a probability space $(\Omega, \mathcal{F}, P)$ and a one-dimensional Wiener process (Brownian motion) $\{W_t\}_{t\in [0,\infty)}$ adapted to its natural filtration. By Wiener process I mean that $\{W_t\}_{t\in [0,\infty)}$ fulfils following properties

$P(\{\omega:W_0(\omega) = 0\}) = 1$
$\forall t_0,t_1,...,t_n \in [0,\infty): t_0 \le t_1 \le ... \le t_n$ we have that $W_{t_0}, W_{t_1}-W_{t_0}, ..., W_{t_n} - W_{t_{n-1}}$ are independent
$\forall s \in [0,t): W_t - W_s \sim N(0,t-s)$
$P(\{\omega: t \mapsto W_t(\omega) - \textrm{continuous}\}) = 1$

Now I consider sequence of random variables defined as follows $Y_n = \max_{s \in [n,n+1]}\frac{\vert W_{n+1}-W_{s} \vert}{n}$. I suspect that distribution of $Y_n$ is the same as $Y_1$ multiplied by $\frac{1}{n}$ but I was not able to formally prove it. My question is how to directly calculate the distribution of $Y_n$, considering that maximum function ranges over a unit interval? I would be grateful for some hints.

edited Aug 22 '23 at 14:14

asked Aug 22 '23 at 09:32

MI00

1

Your $Y_n$ is going to have the same distribution as $\frac1n Y_1$ and $Y_1$ has the same distribution as $\max\limits_{t \in [0,1]}\vert W_{t} \vert$ – Henry Aug 22 '23 at 09:49
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I think this can be useful. – openspace Aug 22 '23 at 09:50
@Henry And does$\textrm{Var}(\vert W_t \vert) < \infty$ for each $t\in [0,1]$ imply $\textrm{Var}(Y_1) < \infty$? – MI00 Aug 22 '23 at 14:10
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@MI97 An answer to a related question suggests that the tail of the distribution is not heavy so I would expect $\textrm{Var}(Y_1)$ to be finite – Henry Aug 22 '23 at 14:24

Distribution of maximum of random variables taken over an uncountable set

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